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[Tutorial M06] Necessary & sufficient conditions

The concepts of necessary and sufficient conditions help us understand and explain the different kinds of connections between concepts, and how different states of affairs are related to each other.

§ M06.1 Necessary conditions

To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y. A necessary condition is sometimes also called "an essential condition". Some examples :

  • Having four sides is necessary for being a square.
  • Being brave is a necessary condition for being a good soldier.
  • Not being divisible by four is essential for being a prime number.

To show that X is not a necessary condition for Y, we simply find a situation where Y is present but X is not. Examples :

  • Being rich is not necessary for being happy, since a poor person can be happy too.
  • Being Chinese is not necessary for being a Hong Kong permanent resident, since a non-Chinese can becoming a permanent resident if he or she has lived in Hong Kong for seven years.

Additional remarks about necessary conditions :

  • We invoke the notion of a necessary condition very often in our daily life, even though we might be using different terms. For example, when we say things like "life requires oxygen", this is equivalent to saying that the presence of oxygen is a necessary condition for the existence of life.
  • A certain state of affairs might have more than one necessary condition. For example, to be a good concert pianist, having good finger techniques is a necessary condition. But this is not enough. Another necessary condition is being good at interpreting piano pieces.

§ M06.2 Sufficient conditions

To say that X is a sufficient condition for Y is to say that the presence of X guarantees the presence of Y. In other words, it is impossible to have X without Y. If X is present, then Y must also be present. Again, some examples :

  • Being a square is sufficient for having four sides.
  • Being divisible by 4 is sufficient for being an even number.

To show that X is not sufficient for Y, we come up with cases where X is present but Y is not. Examples :

  • Loving someone is not sufficient for being loved. A person who loves someone might not be loved by anyone perhaps because she is a very nasty person.
  • Loyalty is not sufficient for honesty because one might have to lie in order to protect the person one is loyal to.

Additional remarks about sufficient conditions :

  • Expressions such as "If X then Y", or "X is enough for Y", can also be understood as saying that X is a sufficient condition for Y.
  • Some state of affairs can have more than one sufficient condition. Being blue is sufficient for being colored, but of course being green, being red are also sufficient for being coloured.

§ M06.3 Four possibilities

Given two conditions X and Y, there are four ways in which they might be related to each other:

  • X is necessary but not sufficient for Y.
  • X is sufficient but not necessary for Y.
  • X is both necessary and sufficient for Y. (or "jointly necessary and sufficient")
  • X is neither necessary nor sufficient for Y.
This classification is very useful in when we want to clarify how two concepts are related to each other. Here are some examples :
  • Having four sides is necessary but not sufficient for being a square (since a rectangle has four sides but it is not a square).
  • Having a son is sufficient but not necessary for being a parent (a parent can have only one daughter).
  • Being an unmarried man is both necessary and sufficient for being a bachelor.
  • Being a tall person is neither necessary nor sufficient for being a successful person.

§ M06.4 Exercises

Question 1.

Rewrite these claims in terms of necessary and / or sufficient conditions :

(a) You must pay if you want to enter.
Payment is necessary for entrance

(b) A cloud chamber is needed to observe subatomic particles.
A cloud chamber is necessary for observing subatomic particles.

(c) If something is an electron it is a charged particle.
Being an electron is sufficient for being a charged particle.

(d) I will pay for lunch if and only if you pay for dinner.
My paying for lunch is necessary and sufficient for your paying for dinner.

Question 2.

Suppose Tom is a tall but unsuccessful person. Does it show that (a) being tall is not sufficient for being successful, or (b) being tall is not necessary for being successful?
Yes for (a) and no for (b).

Question 3.

Discuss how these conditions are related to each other and explain your answers :

(a) not being poor, being rich
The first is necessary but not sufficient for the second, since it is possible to be neither poor nor rich.

(b) being an even number, being divisible by 2
The first is both necessary and sufficient for the second.

(c) being an intelligent student, being the most intelligent student
Neither necessary nor sufficient. Why?

(d) having ten dollars, having more than five dollars
The first is sufficient but not necessary for the second.

(e) the presence of the rule of law, being a just society
This is a difficult one!

(f) giving money to another person in exchange for a favour, corruption

(g) taking place on a weekday, not being held on Saturday

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